Optimal controls related to microalgae cultivation

By Ignacio Fierro

In recent years, there has been a notable increase in the industrial cultivation of microalgae, accompanied by the development of new technologies such as photobioreactors, sensors, and controllers. These new technologies are capable of estimating biomass inside the photobioreactor. When the data is recorded, mathematical models can be fitted in order to forecast the biomass taking into account different factors, as light intensity, dilution rate, pH, nutrient concentrations, etc. In industrial production levels, several questions are of interest in order to make the most of our resources and constraints. For example: What is the optimal performance of the system involved? How should the photobioreactor be operated to maximize the production of biomass or any compound of interest produced by the microalgae? Which operational parameters should be constant? Do some biological phenomena specific to microalgae influence production?

Accurate mathematical models describing biomass evolution in a photobioreactor play a crucial role in addressing these questions. However, mathematics cannot answer all the questions, especially when a factor is not considered, or its influence is not well stipulated, or worse, when two factors together act differently than when they act separately. Modeling the behavior of biological organisms present an additional challenge: acclimatization and adaptation. Microorganisms, in particular microalgae, are able to acclimate to certain external factors, e.g., they can adjust the amount and composition of pigments depending on the amount of light they perceive. This acclimatization can impact growth in the photobioreactor and subsequently affect the model’s prediction.

But let’s forget, for the moment, the limitations of the models and let’s assume that we have one that describes very well the biomass evolution in our photobioreactor which can be operated in continuous mode, considering the influence of light and dilution rate. While other factors are kept under control, such as pH and temperature, other factors are not assumed to be limiting, as nutrients (they have enough nutrients for growth). We now are interested to study the best performance of our system using some mathematics.

Before getting into this topic, a definition is needed. Control theory is a branch of engineering and mathematics that deals with adjusting a system’s behavior to achieve desired outcomes. This manipulation is called control. In the context of a photobioreactor, controls may include light intensity, nutrient inlet concentration, temperature, pH, medium volume, CO2, or dilution rate. The questions posed above can be formulated in the optimal control language.

First, define the objectives. These may include maximizing biomass production, optimizing nutrient utilization, or enhancing the production of specific compounds. Second, the key controls or operational parameters should be determined, for instance, pH is maintained at constant levels injecting CO2 to ensure the survival of the species concerned, but light intensity can be controlled depending on the concentration of the biomass to avoid photoinhibition and trigger the photosynthetic machinery of the culture. These factors vary depending on how the microalgae are being cultivated. In addition to this, to study an optimal control problem, we must also consider the limitations of the model. For example, if our model does not consider the influence of nutrients, we will not be able to consider this factor as a control, it just makes no sense.

Then, given a certain objective, for example, to maximize biomass production, the question is: how do we set our controls? The Pontryagin’s maximum principle (PMP) is a mathematical tool that can be used to answer these new questions. When a mathematical model of our system, written as a set of ordinary differential equations, has well-defined the influence of these controls, PMP can be applied to find the optimal control. The principle provides the necessary conditions for a control to be optimal in the presence of constraints. However, these necessary conditions are not easy to decipher and most of the time one must resort to a software that is able to solve an approximation of the problem. By combining the theoretical results delivered by PMP and the numerical simulations of some software, we can gradually discover the structure of our optimal control. Why gradually? Discovering the secrets of our unknown optimal control is like piecing together a puzzle. At the beginning, a few conditions can be derived from the PMP, for example, the optimal operation of the dilution rate can take three values, zero, at its maximum value and a third value that’s unknown. Numerical simulations can help us determine this unknown value.

The engineering challenge does not end after finding the solution to our control problem. This solution has to be evaluated and interpreted. Perhaps the solution is outside the margins in which our model was tested and we are not sure if the predictions will be as expected, since the photobioreactor has never worked in those conditions, or even worse, it is not possible for the photobioreactor to work in these conditions.

In conclusion, control theory emerges as a valuable framework to address questions related to optimizing system performance in industrial-scale microalgae cultivation. By formulating optimal control problems and leveraging mathematical tools like the Pontryagin’s maximum principle (PMP), researchers can navigate the intricate relationships between operational parameters and desired outcomes. However, the applying PMP requires careful interpretation. This often involves using numerical simulations to uncover optimal control strategies.

To understand an optimal control problem in a microalgae-based bioprocess it is crucial to formulate the system on differential equations, pinpointing the controls in order to apply the Pontryagin maximum principle optimality conditions and utilize a numerical solver.